Magic without a phase: phase-independent stabilizer Rényi entropy in gluon scattering

Abstract: Magic, also known as non-stabilizerness, measures the usefulness of a quantum state for quantum computation. While magic is defined relative to a choice of computational basis, in some physical settings the available data determine this basis only up to local phase conventions. In this paper, we generalize the notion of magic and formulate it in a phase-independent manner, and hence define a generalized stabilizer Rényi entropy. As a case study, we consider higher-multiplicity tree-level gluon scattering, interpreting the outgoing helicities as qubits. In this setting, the helicity data naturally determine a local basis for each qubit but leave a phase ambiguity. For $3\to 2$ scattering, we find that the final-state phase-independent magic is generically larger than the maximum attainable in $2\to 2$ scattering. For $2 \to 3$ scattering, we find a nonzero minimal value approached in the soft limit. Moreover, when the three outgoing momenta become symmetric, the magic approaches a local minimum only a few percent above the soft-limit value. In all cases considered, the color dependence cancels from the phase-independent stabilizer Rényi entropy.

• The paper defines a phase-independent stabilizer Rényi entropy by averaging over local U(1) basis rephasings, removing convention-dependent magic while retaining sensitivity to non-stabilizerness.
• The paper finds that 3→2 gluon scattering reaches M₂ = ln(27/13) ≈ 0.731, well above the 2→2 maximum of ln(7/5) ≈ 0.336, with complex helicity phases driving the enhancement.
• The paper shows that 2→3 scattering has a lower ceiling of about 0.38, while color dependence cancels in the studied amplitudes, making the phase-independent entropy a kinematics-only diagnostic with open experimental and theoretical limitations.

Motivation and central construction

The stabilizer Rényi entropy (SRE) quantifies non-stabilizerness ("magic"), the resource that distinguishes classically simulable Clifford dynamics from genuinely quantum-computational states. The paper's starting point is that the SRE, while Clifford-invariant, is not invariant under generic local unitaries; in particular, a local diagonal rephasing $|+\rangle_i \to e^{i\theta_i}|+\rangle_i$ changes the assigned magic continuously. A Bell state in one phase frame can acquire nonzero SRE in another. Since physical data — helicity amplitudes in particular — fix the computational basis only up to such phase conventions, the authors construct a phase-independent generalization: the stabilizer Rényi entropy averaged over the $U(1)^k$ orbit of basis rephasings, evaluated inside the logarithm (2607.13134). They contrast this "typical-phase" invariant with the minimization-over-phases prescription used in non-local magic, noting that the two are distinct and that neither is uniquely canonical.

For a two-coefficient state $r|+-\rangle + |-+\rangle$, the phase-averaged quantity is computed in closed form, $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$, and shown numerically to coincide with the exact orbit average; by Jensen's inequality it is a rigorous lower bound on the true average. The rephasing lifts the degeneracy between the trivial state ($r=0$, zero magic) and the Bell state ($r=1$, $\widetilde{M}_2 = \ln(8/7)\approx 0.134$), the latter being generically non-stabilizer once phases are unresolved. The maximum is $\ln(7/5)\approx 0.336$ at $r = (\sqrt{7}-\sqrt{3})/2$.

Gluons as qubits

The application treats tree-level gluon scattering: each outgoing gluon helicity is a qubit, and the color-dressed amplitude, with external colors and momenta fixed, is the polarization wavefunction prepared by the $S$-matrix. Using the DDM color decomposition and the Parke–Taylor formula for MHV and anti-MHV partial amplitudes, the authors extract the helicity coefficients for three processes. A recurring structural result is that the color dependence cancels from the phase-independent SRE: for the MHV/anti-MHV structures considered, ratios of coefficient moduli reduce to ratios of spinor-product magnitudes, so $U(1)^k$0 is a function of kinematics alone. The authors state this cancellation is established only for the amplitudes they treat and explicitly leave its general validity open.

Three scattering processes

For $U(1)^k$1 scattering with incoming helicity $U(1)^k$2, the final state is a two-coefficient family with $U(1)^k$3. The phase-independent SRE vanishes at $U(1)^k$4, peaks at $U(1)^k$5 near $U(1)^k$6 and $U(1)^k$7, and dips to $U(1)^k$8 at $U(1)^k$9 (the Bell point). The invariant also removes spurious wiggles that appear in the bare $r|+-\rangle + |-+\rangle$0 when a particular color configuration is chosen, confirming those artifacts are pure phase-convention effects.

For $r|+-\rangle + |-+\rangle$1 scattering with incoming helicity $r|+-\rangle + |-+\rangle$2, the final state acquires a third component $r|+-\rangle + |-+\rangle$3 with genuinely complex coefficients inherited from the Parke–Taylor and anti-MHV structures. The phase-independent SRE never vanishes and reaches $r|+-\rangle + |-+\rangle$4, generically exceeding anything attainable in $r|+-\rangle + |-+\rangle$5 scattering and approaching the conjectured two-qubit maximum $r|+-\rangle + |-+\rangle$6. At the symmetric kinematic point $r|+-\rangle + |-+\rangle$7, the bare $r|+-\rangle + |-+\rangle$8 reaches $r|+-\rangle + |-+\rangle$9, only slightly below the conjectured bound. The authors attribute this enhancement to the complex phases, which a real-coefficient submanifold cannot supply.

For $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$0 scattering — the paper's first three-qubit case — the pattern inverts. The phase-independent SRE is bounded above by roughly $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$1, below the typical values in the $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$2 case, so additional outgoing legs do not increase magic. It is bounded below by a floor of $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$3, approached in the soft limit $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$4. Notably, when the three outgoing energies become equal ($\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$5, $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$6), the magic settles at $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$7, a local minimum only a few percent above the soft-limit floor.

Process	Range of $\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$8	Extremal values
$\widetilde{M}_2 = \ln[(1+r^2)^4/(1+12r^4+r^8)]$9	$r=0$0 to $r=0$1	max $r=0$2; local min $r=0$3
$r=0$4	never vanishes	max $r=0$5
$r=0$6	$r=0$7 to $r=0$8	floor in soft limit; symmetric point $r=0$9

Limitations and open questions

Several qualifications are stated in the paper. The averaging prescription replaces the orbit average by averaging inside the logarithm; the authors argue this is accurate when phase-dependent terms are small relative to the constant part but note that whether this approximation is generally valid is left to future work. The color cancellation is proven only for the MHV/anti-MHV five-point amplitudes considered. Whether $r=1$0 remains a genuine resource monotone, as the bare SRE does for $r=1$1, is unresolved, as is the question of whether the averaged or minimized phase-independent quantity is the physically appropriate invariant. Finally, connecting $r=1$2 to collider observables faces the practical obstacles of gluon polarimetry, color decoherence, and hadronization; the paper leaves open how the phase-independent SRE can be measured experimentally, in contrast to the top-quark magic measurements at the LHC.

Conclusion

The paper defines a phase-averaged, $r=1$3-invariant stabilizer Rényi entropy and applies it to tree-level gluon scattering at four- and five-point multiplicity. The main quantitative findings are that $r=1$4 scattering produces phase-independent magic ($r=1$5) far above the $r=1$6 ceiling ($r=1$7), that $r=1$8 scattering saturates at a lower ceiling despite the larger Hilbert space, and that color dependence cancels in all cases computed, leaving a kinematics-only observable. The construction provides a basis-convention-independent diagnostic of non-stabilizerness for any system whose data determine the computational basis only up to local phases.